tf.einsum

Tensor contraction over specified indices and outer product.

View aliases

Main aliases

tf.linalg.einsum

Compat aliases for migration

See Migration guide for more details.

tf.compat.v1.einsum, tf.compat.v1.linalg.einsum, `tf.compat.v2.einsum`, `tf.compat.v2.linalg.einsum`

tf.einsum(
    equation, *inputs, **kwargs
)

This function returns a tensor whose elements are defined by equation, which is written in a shorthand form inspired by the Einstein summation convention. As an example, consider multiplying two matrices A and B to form a matrix C. The elements of C are given by:

  C[i,k] = sum_j A[i,j] * B[j,k]

The corresponding equation is:

  ij,jk->ik

In general, the equation is obtained from the more familiar element-wise equation by

1. removing variable names, brackets, and commas,
2. replacing "*" with ",",
3. dropping summation signs, and
4. moving the output to the right, and replacing "=" with "->".

Many common operations can be expressed in this way. For example:

# Matrix multiplication
>>> einsum('ij,jk->ik', m0, m1)  # output[i,k] = sum_j m0[i,j] * m1[j, k]

# Dot product
>>> einsum('i,i->', u, v)  # output = sum_i u[i]*v[i]

# Outer product
>>> einsum('i,j->ij', u, v)  # output[i,j] = u[i]*v[j]

# Transpose
>>> einsum('ij->ji', m)  # output[j,i] = m[i,j]

# Trace
>>> einsum('ii', m)  # output[j,i] = trace(m) = sum_i m[i, i]

# Batch matrix multiplication
>>> einsum('aij,ajk->aik', s, t)  # out[a,i,k] = sum_j s[a,i,j] * t[a, j, k]

To enable and control broadcasting, use an ellipsis. For example, to do batch matrix multiplication, you could use:

einsum('...ij,...jk->...ik', u, v)

This function behaves like numpy.einsum, but does not support:

• Subscripts where an axis appears more than once for a single input (e.g. ijj,k->ik) unless it is a trace (e.g. ijji).

Args
equation	a str describing the contraction, in the same format as numpy.einsum.
*inputs	the inputs to contract (each one a Tensor), whose shapes should be consistent with equation.
name	A name for the operation (optional).

Returns
The contracted Tensor, with shape determined by equation.

Raises
ValueError	If the format of equation is incorrect, the number of inputs implied by equation does not match len(inputs), an axis appears in the output subscripts but not in any of the inputs, the number of dimensions of an input differs from the number of indices in its subscript, or the input shapes are inconsistent along a particular axis.